Research On Controllability Index Of Complex Networks

At present,the research on complex networks mostly considers the selection of driving nodes,and makes the network reach the expected control state by selecting the least driving nodes.Because there is more than one set of driving nodes,the controllability index of the network under different control schemes is different.For a large-scale network system,how to select the driving node to make it controllable,and the controllability index under the premise of controllability is as small as possible is the key of the research.A small controllability index means that the large-scale network system can reach the stable state faster,and it is of great significance in practical application.Starting from the traditional control theory,this thesis introduces the knowledge of graph theory to describe the real network,judges the control performance of a complex network through the structural controllability index K,and puts forward the necessary and sufficient conditions for K controllability,which lays a theoretical foundation for the research of large-scale networks in this thesis.Firstly,the complex directed network is studied from the perspective of control chain.By decomposing the complex network into independent and controllable control chains,the complexity of network driven node analysis is simplified.For a directed network without driving nodes,the directed network is decomposed into multiple control chains by studying the node degree and deleting edges,and the root node of the control chain is taken as the driving node set of the whole network.At the same time,the structural controllability index k is introduced to study the control chain length.Finally,based on the node degree,this thesis proposes a blocking algorithm for the network system matrix to obtain the driving node set of the network and the corresponding structural controllability index K.Simulation results on large-scale networks verify the effectiveness of the algorithm.Secondly,the thesis studies the cactus structure based on the chain controllable structure.In the sense of controllability,compared with the chain structure,large-scale networks can usually be formed by fewer cactus structures,which means fewer driving nodes.By constructing an effective network system structure matrix decomposition algorithm,this thesis decomposes the large-scale network system into independent and controllable sub cactus system,and forms the selection scheme of driving node set.At the same time,combined with the concept of structural controllability index K,the necessary and sufficient conditions for K controllability of large-scale network system are proposed,and the relevant theorems are verified by simulation.Based on the above research,this thesis extends the results of k-structure controllability of directed network to undirected network,proposes the structure controllable tree of undirected network,and proves its structure controllable.Finally,by studying the relationship between the value of controllability index K and the number of control nodes of the network,and discussing the performance of complex network,this thesis discusses how to select the value of control nodes and controllability index K to make the network performance better.At the same time,the research is extended to undirected networks,the structure controllable tree of undirected networks is proposed,and its structure is proved to be controllable.